We analyze generalized space-time fractional motions on undirected networks and lattices. The continuoustime random walk (CTRW) approach of Montroll and Weiss is employed to subordinate a space fractional walk to a generalization of the time-fractional Poisson renewal process. This process introduces a non-Markovian walk with long-time memory effects and fat-tailed characteristics in the waiting time density. We analyze 'generalized space-time fractional diffusion' in the infinite d -dimensional integer lattice Z d . We obtain in the diffusion limit a 'macroscopic' space-time fractional diffusion equation. Classical CTRW models such as with Laskin's fractional Poisson process and standard Poisson process which occur as special cases are also analyzed. The developed generalized space-time fractional CTRW model contains a four-dimensional parameter space and offers therefore a great flexibility to describe real-world situations in complex systems. of the mean-square displacement in anomalous diffusion [5]. It has been demonstrated that such anomalous diffusive behavior is well described by a random walk subordinated to the fractional generalization of the Poisson process. This process which was to our knowledge first introduced by Repin and Saichev [13], was developed and analyzed by Laskin who called this process the 'fractional Poisson process' [14,15]. The Laskin's fractional Poisson process was further generalized in order to obtain greater flexibility to adopt real-world situations [16,17,18]. We refer this renewal process to as 'generalized fractional Poisson process' (GFPP). Recently we developed a CTRW model of a normal random walk subordinated to a GFPP [17,18].The purpose of the present paper is to explore space fractional random walks that are subordinated to a GFPP. We analyze such motions in undirected networks and as a special application in the multidimensional infinite integer lattice Z d .